Continuous normalizing flows (CNFs) learn the probability path between a reference distribution and a target distribution by modeling the vector field generating said path using neural networks. Recently, Lipman et al. (2022) introduced a simple and inexpensive method for training CNFs in generative modeling, termed flow matching (FM). In this paper, we repurpose this method for probabilistic inference by incorporating Markovian sampling methods in evaluating the FM objective, and using the learned CNF to improve Monte Carlo sampling. Specifically, we propose an adaptive Markov chain Monte Carlo (MCMC) algorithm, which combines a local Markov transition kernel with a non-local, flow-informed transition kernel, defined using a CNF. This CNF is adapted on-the-fly using samples from the Markov chain, which are used to specify the probability path for the FM objective. Our method also includes an adaptive tempering mechanism that allows the discovery of multiple modes in the target distribution. Under mild assumptions, we establish convergence of our method to a local optimum of the FM objective.

We are interested in learning generative models for complex geometries described via manifolds, such as spheres, tori, and other implicit surfaces. Current extensions of existing (Euclidean) generative models are restricted to specific geometries and typically suffer from high computational costs. We introduce Moser Flow (MF), a new class of generative models within the family of continuous normalizing flows (CNF). MF also produces a CNF via a solution to the change-of-variable formula, however differently from other CNF methods, its model (learned) density is parameterized as the source (prior) density minus the divergence of a neural network (NN). The divergence is a local, linear differential operator, easy to approximate and calculate on manifolds. Therefore, unlike other CNFs, MF does not require invoking or backpropagating through an ODE solver during training.

The placement of Cloud-Native Network Functions (CNFs) across the Cloud-Continuum represents a core challenge in the orchestration of current 5G and future 6G networks. The process involves the placement of interdependent computing tasks, structured as Service Function Chains, over distributed cloud infrastructures. This is achieved while satisfying strict resource, bandwidth and latency constraints. It is acknowledged that classical approaches, including mixed-integer nonlinear programming, heuristics and reinforcement learning are limited in terms of scalability, constraint handling and generalisation capacity. In the present study, a novel theoretical framework is proposed, which is based on Denoising Diffusion Probabilistic Models (DDPM) for CNF placement. The present approach proposes a reconceptualisation of placement as a generative graph to assignment task, where the placement problem is encoded as a heterogeneous graph, and a Graph Neural Network denoiser is trained to iteratively refine noisy CNF-to-cloud assignment matrices. The model incorporates constraint-specific losses directly into the loss function, thereby allowing it to learn feasible solution spaces. The integration of the DDPM formulation with structured combinatorial constraints is achieved through a rigorous and systematic approach. Extensive evaluations across diverse topologies have been conducted, which have confirmed that the model consistently produces feasible solutions with orders of magnitude faster inference than MINLP solvers. The results obtained demonstrate the potential of diffusion-based generative modelling for constrained network embedding problems, making an impact towards the practical, scalable orchestration of distributed Cloud-Native Network Functions.

Continuous normalizing flows (CNFs) learn the probability path between a reference distribution and a target distribution by modeling the vector field generating said path using neural networks. Recently, Lipman et al. (2022) introduced a simple and inexpensive method for training CNFs in generative modeling, termed flow matching (FM). In this paper, we repurpose this method for probabilistic inference by incorporating Markovian sampling methods in evaluating the FM objective, and using the learned CNF to improve Monte Carlo sampling. Specifically, we propose an adaptive Markov chain Monte Carlo (MCMC) algorithm, which combines a local Markov transition kernel with a non-local, flow-informed transition kernel, defined using a CNF. This CNF is adapted on-the-fly using samples from the Markov chain, which are used to specify the probability path for the FM objective.

Estimating physical parameters from data is a crucial application of machine learning (ML) in the physical sciences. However, systematic uncertainties, such as detector miscalibration, induce data distribution distortions that can erode statistical precision. In both high-energy physics (HEP) and broader ML contexts, achieving uncertainty-aware parameter estimation under these domain shifts remains an open problem. In this work, we address this challenge of uncertainty-aware parameter estimation for a broad set of tasks critical for HEP. We introduce a novel approach based on Contrastive Normalizing Flows (CNFs), which achieves top performance on the HiggsML Uncertainty Challenge dataset. Building on the insight that a binary classifier can approximate the model parameter likelihood ratio, we address the practical limitations of expressivity and the high cost of simulating high-dimensional parameter grids by embedding data and parameters in a learned CNF mapping. This mapping yields a tunable contrastive distribution that enables robust classification under shifted data distributions. Through a combination of theoretical analysis and empirical evaluations, we demonstrate that CNFs, when coupled with a classifier and established frequentist techniques, provide principled parameter estimation and uncertainty quantification through classification that is robust to data distribution distortions.

We are interested in learning generative models for complex geometries described via manifolds, such as spheres, tori, and other implicit surfaces. Current extensions of existing (Euclidean) generative models are restricted to specific geometries and typically suffer from high computational costs. We introduce Moser Flow (MF), a new class of generative models within the family of continuous normalizing flows (CNF). MF also produces a CNF via a solution to the change-of-variable formula, however differently from other CNF methods, its model (learned) density is parameterized as the source (prior) density minus the divergence of a neural network (NN). The divergence is a local, linear differential operator, easy to approximate and calculate on manifolds.

This paper introduces a method for incorporating prior knowledge encoded as logical rules to improve the performance of deep learning models. In particular, it takes logical rules which are in decomposable and deterministic negation normal form (d-DNNF), and proposes using an augmented graph convolution network to embed them into a vector space. This embedding is then regularised according to the logical constraints, allowing the addition of a "logic loss" term to train models obeying these logical rules. Incorporating (symbolic) background knowledge to improve performance of deep learning methods is an interesting and valuable direction, and from the experiments using a d-DNNF rather than a CNF appears to be beneficial. However, for me the notion of using a d-DNNF as the source of background knowledge raises a few issues which I feel are not addressed in the paper.

Continuous normalizing flows (CNFs) can model data distributions with expressive infinite-length architectures. But this modeling involves computationally expensive process of solving an ordinary differential equation (ODE) during maximum likelihood training. Recently proposed flow matching (FM) framework allows to substantially simplify the training phase using a regression objective with the interpolated forward vector field. In this paper, we propose an interpolant-free dual flow matching (DFM) approach without explicit assumptions about the modeled vector field. DFM optimizes the forward and, additionally, a reverse vector field model using a novel objective that facilitates bijectivity of the forward and reverse transformations. Our experiments with the SMAP unsupervised anomaly detection show advantages of DFM when compared to the CNF trained with either maximum likelihood or FM objectives with the state-of-the-art performance metrics.

Language Modelling has been a central part of Natural Language Processing for a very long time and in the past few years LSTM-based language models have been the go-to method for commercial language modeling. Recently, it has been shown that when looking at language modelling from a matrix factorization point of view, the final Softmax layer limits the expressiveness of the model, by putting an upper bound on the rank of the resulting matrix. Additionally, a new family of neural networks based called NeuralODEs, has been introduced as a continuous alternative to Residual Networks. Moreover, it has been shown that there is a connection between these models and Normalizing Flows. In this work we propose a new family of language models based on NeuralODEs and the continuous analogue of Normalizing Flows and manage to improve on some of the baselines.

Continuous normalizing flows (CNFs) are a generative method for learning probability distributions, which is based on ordinary differential equations. This method has shown remarkable empirical success across various applications, including large-scale image synthesis, protein structure prediction, and molecule generation. In this work, we study the theoretical properties of CNFs with linear interpolation in learning probability distributions from a finite random sample, using a flow matching objective function. We establish non-asymptotic error bounds for the distribution estimator based on CNFs, in terms of the Wasserstein-2 distance. The key assumption in our analysis is that the target distribution satisfies one of the following three conditions: it either has a bounded support, is strongly log-concave, or is a finite or infinite mixture of Gaussian distributions. We present a convergence analysis framework that encompasses the error due to velocity estimation, the discretization error, and the early stopping error. A key step in our analysis involves establishing the regularity properties of the velocity field and its estimator for CNFs constructed with linear interpolation. This necessitates the development of uniform error bounds with Lipschitz regularity control of deep ReLU networks that approximate the Lipschitz function class, which could be of independent interest. Our nonparametric convergence analysis offers theoretical guarantees for using CNFs to learn probability distributions from a finite random sample.